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MATH237501 This question paper consists of 4 printed pages, each of which is identified by the reference MATH237501. All calculators must carry an approval sticker issued by the School of Mathematics. c University of Leeds School of Mathematics May/June 2016 MATH237501 Linear Differential Equations ...

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MATH237501

This question paper consists of 4 printed pages, each of which is identified by the reference MATH237501.

All calculators must carry an approval sticker issued by the School of Mathematics.

University c of Leeds School of Mathematics May/June 2016 MATH237501 Linear Differential Equations and Transforms Time Allowed: 2 21 hours Answer no more than 4 questions. If you attempt 5, only the best 4 will be counted. All questions carry equal marks.

1

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MATH237501 1. (a) Consider the differential equation for y(x): y ′′ + p(x)y ′ + q(x)y = 0.

(1)

Defining the Wronskian W (x) of two solutions of (1) by W (x) = y1 y2′ − y ′1 y2 , show that W ′ + p(x)W = 0, and that

d dx

y2 y1

!

=

W (x) . y12

Suppose y1 and y2 are linearly independent and that the general solution is written as y = c1 y1 + c2 y2 . If y(x0 ) = α, y ′ (x0 ) = β, derive a formula for c1 , c2 in terms of α, β, yi (x0 ), yi′(x0 ), showing clearly the role of W (x0 ). (b) Show that y1 = x sin x satisfies the equation x2 y ′′ − 2xy ′ + (x2 + 2)y = 0. Use the formulae of Part (a) to find W (x) in this case and hence to find a second independent solution y2 (x). d cot x = −cosec2 x useful.] [Hint: You may find dx 2. (a) For the equation S(x)y ′′ + P (x)y ′ + Q(x)y = 0,

S, P, Q polynomials,

define the meaning of an ordinary and a singular point. If x0 is a singular point, give the conditions for it to be regular. What form of series solution would you seek about i. an ordinary point? ii. a regular singular point? (b) Show that x = 0 is a regular singular point for the equation x2 y ′′ + 4xy ′ + (2 − x2 )y = 0, and show that the indicial equation has roots r1 and r2 which differ by an integer. Consider a Frobenius solution of this equation of the form y = x−2

∞ X

n=0

an xn ,

a0 6= 0,

and show that it contains two arbitrary coefficients. Use the general recurrence relation to derive the general term in each series. What elementary functions are defined by these series?

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MATH237501 3. (a) Legendre’s polynomials can be constructed via a generating function, with G(x, t) = (1 − 2tx + t2 )−1/2 =

∞ X

Pn (x)tn .

n=0

Use the binomial expansion 1 3 5 3 τ + τ2 − τ + ..., 2 8 16

(1 + τ )−1/2 = 1 −

to calculate the four polynomials P0 , . . . , P3 . Use the recurrence relation (n + 1)Pn+1 = (2n + 1)xPn − nPn−1 , to find P4 . (b) Defining hf, gi =

Z 1

f (x)g(x)dx,

−1

you may assume that hPm , Pn i = 0,

when m 6= n.

Use the recurrence relation to show that hPn , Pn i =



2n − 1 hPn−1 , Pn−1 i . 2n + 1 

Show by direct calculation that hP0 , P0 i = 2 and hence deduce that hPn , Pn i = Find a0 , a1 in the expansion ex =

∞ X

2 . 2n + 1

an Pn .

n=0

4. Consider the following initial boundary value problem for the heat equation: ut = uxx

for 0 < x < 1, t > 0,

u(0, t) = u(1, t) = 0,

u(x, 0) = ϕ(x).

(a) Use the method of separation of variables to show that the general form of the solution satisfying the boundary conditions is u(x, t) =

∞ X

cn e−n

2 2

π t

sin(nπx).

n=1

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MATH237501 (b) Use the trigonometric identities 1 (cos(A − B) − cos(A + B )) 2 1 sin2 A = (1 − cos 2A), 2

sin A sin B =

to show the orthogonality relations Z 1 0

1 sin(mπx) sin(nπx) dx = δmn = 2

(

0 if m 6= n, if m = n,

1 2

where m and n are positive integers. (c) Hence find the coefficients cn for the explicit case: ϕ(x) = x for 0 < x < 1. 5. The Fourier Transform of a piecewise smooth function f (x), for which exists and is finite, is given by 1 F[f (x)] = √ 2π

Z ∞

R∞

−∞

|f (x)|dx

f (x)e−ikx dx.

−∞

(a) Show that F [f (x − a)] = e−iak F [f (x)]. (b) Show that if f (x) is a differentiable function, with lim f (x) = 0,

x→±∞

then F [f ′ (x)] = ik F [f (x)]. (c) Consider the partial differential equation 3

∂u ∂u + 2t = 0, ∂t ∂x

subject to initial condition u(x, 0) = f (x). Defining uˆ(k, t) = F[u(x, t)] and fˆ(k) = F[f (x)], show that uˆ(k, t) = A(k)e−

ikt2 3

,

for some function A(k), to be determined. Hence use part (a) to deduce the form of u(x, t). If ( 1 if 1 < x < 2, f (x) = 0 elsewhere, sketch the graph of u(x, t) for t = 0, t = 2, t = 3? Deduce that this is a travelling wave, moving to the right. What is its speed?

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